Proving this theorem was a class assignment. My proof may or may not be original (probably not).
Theorem (
BONAPARTE)
Let
ABC be a triangle.
Erect
equilateral triangles A'BC,
AB'C,
ABC' so either all overlap with Δ
ABC or none overlap, with
centroids a,
b,
c respectively. Then Δ
abc is an
equilateral triangle (known as Napoleon's triangle).
_.B'
_ ./ |
_ ./ /
_ ./ |
_ ./ /
C'________________A / |
\ /\ .b /
\ / \ |
\ .c / \ /
\ / \ |
\ / \ /
\ / \ |
\ / \ /
\ / \ |
\/________________________\/
B \ /C
\ /
\ /
\ /
\ .a /
\ /
\ /
\ /
\ /
\ /
\ /
\ /
\/
A'
Proof
Make two comparisons between Δ
cAb and Δ
BAB' to see that they are
similar triangles:
∠ cAb = ∠ cAB + ∠ BAC + ∠ CAb
= π/6 + ∠ BAC + π/6
= ∠ BAC + π/3
∠ BAB' = ∠ BAC + ∠ CAB'
= ∠ BAC + π/3
(same angles)
By the fixed ratios of lengths in an equilateral triangle:
|c - A| = |B - A| / sqrt(3)
|b - A| = |B' - A| / sqrt(3)
∴
|c - A| / |b - A| = |B - A| / |B' - A|
(preserved ratio of side lengths)
They are similar triangles. Hence the third length must have the same ratio:
|c - b| = |B' - B| / sqrt(3)
By reasons of symmetry,
|a - b| = |B' - B| / sqrt(3)
Hence
|
a - b| = |
b - c|
and by reasons of symmetry, all sides of
Δ
abc are of equal length.
This proof works for the case where the constructed equilateral triangles point inward, too.
QED
Napoleon Points
Lines
laA ,
lbB , and
lcC meet at the first Napoleon point when the three equilateral triangles point away from Δ
ABC. When they point toward Δ
ABC, the lines meet at a different point known as the second Napoleon point.
Fermat Point
The point
P of a triangle
ABC that minimizes the sum of distances |
A - P| + |
B - P| + |
C - P| is known as the Fermat point.
When each of the three angles of Δ
ABC are less than 120°, the Fermat point is the
concurrent point of lines
lA'A,
lB'B, and
lC'C. It is also where the
circumcircles of the erected
equilateral triangles meet. When one of the
vertices has an angle greater than 120
°, then that
vertex is the Fermat point.
The Fermat point was not discovered by Fermat himself. Instead, he was the one who asked where such a point would exist, to which Toricelli was the first to answer this question.